Electromagnetic radiation, such as light, may be concentrated through an imaging optical system. The most familiar of such systems is a burning glass in which sunlight is focused onto a flammable surface, such as wood, by a magnifying glass. Curiously enough, elementary geometrical optics does not yield a theoretical limit on concentration for image-forming systems of this kind. According to the usual paraxial or Gaussian optics theory, when an object of linear dimension .eta. is imaged at size .eta.', the convergence angles are in the inverse ratio of these dimensions. FIG. 1A shows this relationship with media of different refractive indices in the source and absorber spaces. According to classical optics, the Lagrange invariant for this system is given by EQU n.alpha..eta.=n'.alpha.'.eta.'. (1)
For a source at infinity, as in FIG. 1B, the Lagrange invariant takes the form EQU -nh.beta.=n'.alpha.'.eta.'. (2)
If, in FIG. 1B, the concentration ratio is taken as the area of the lens divided by the area of the absorber, then ##EQU1## and a similar expression holds for the case of FIG. 1A with the object at a finite distance. These expressions are based on paraxial optics, i.e., the small-angle approximation, and there is nothing in them which predicts what would happen if the angle .alpha.' is increased for the purpose of increasing concentration. From aberration theory, a lens must be corrected for spherical aberration and chromatic aberration on axis, and the Abbe sine condition also must be satisfied to form a good image of an extended object. Thus, the Abbe condition EQU h=f sin .alpha.', (4)
where f is the paraxial focal length, must be assured for all rays in the system of FIG. 1B.
It is well known to skilled optical designers that ordinary imaging systems with a finite field of view and good aberration correction have not been designed with the image side convergence angle (.alpha.') greater than about 60.degree.. Therefore, the concentration is only about one half of what it could be if the convergence angle were 90.degree.. Examples of systems having large convergence angles are microscope objectives and Abbe condensers. Another example of an imaging system which is less than ideal as a concentrator is the concave paraboloidal mirror which is often used for collecting solar energy. If the paraboloid is used with one infinite conjugate, a large negative coma is developed and spreads the light flux away from the axis, thus decreasing the concentration below that of an ideal, aberration-free system.
Imaging optical systems are fundamentally inefficient light concentrators because such systems impose constraints on the behavior of the ray pattern as described as follows:
An imaging optical system may be mathematically described in terms of a topological transformation (.GAMMA.) between the object space (X) and its image space (.gamma.). EQU .gamma.=.GAMMA.(X)
The fact that .GAMMA. exists implies that for every point in X there exists only one point in .gamma.. Furthermore, .GAMMA. is required to observe the following necessary conditions.
(a) .GAMMA. is a homeomorphism (i.e. it is a continuous transformation for which a continuous inverse exists). From this it follows that .GAMMA. is also one-to-one. PA0 (b) The image .gamma. is essentially aberration free and observes the Abbe Sine condition.
An imaging optical system applied to the problem of radiation concentration is constrained by the imaging requirements ((a) and (b) which mathematically impose restrictions on the permissible choices for the design degrees of freedom). These restrictions only permit a subset of systems which may therefore not be capable of achieving the best performance.
The Lagrange invariant can be regarded as a form of what is sometimes called the radiance theorem (radiance is power per unit projected area per unit solid angle integrated over an appropriate wavelength or frequency interval). In the prior art, the radiance theorem has been interpreted to imply that no optical system passing a single elemental beam can produce an image of a source with greater radiance than the source itself, apart from a factor involving the refractive indices in the source and output spaces.
The Lagrange theorem has also formed the basis for the development of the prior art concept of non-imaging optics. Since the object of such systems is to achieve concentration, whether for solar energy or for other applications, some of the requirements for image-forming systems can be relaxed. Light concentrating systems may contain elements other than ordinary lenses and mirrors (e.g., diffractive or holographic elements), can have aberrations, and need not even form an image at all. The only requirement is that as much as possible of the radiant flux entering one aperture emerges from another aperture of the system. Therefore, non-imaging concentrators are typically better at concentrating light than image-forming systems.
In analyzing imaging light concentrating systems, radiance of the source is considered in terms of elementary beams comprising elementary pencils of light subtending infinitesimal solid angles utilizing classical ray-tracing techniques. In all such cases the maximum achievable concentration ratio, i.e. the ratio of luminance perceived by a target to the luminance of the source, is less than one; in other words, the radiance at the target cannot exceed the radiance of the source. See Giles V. Klein, Wiley and Sons, Inc., "Optics", at pages 132-134, (1970). See also, for example, an analysis of an ordinary search light in Klein at pages 135-138.
A non-imaging light concentrating system may be described as similar to a funnel in which light entering the system over a large area is eventually directed to pass through a much smaller area. In the process, no image of the light source is formed. Non-imaging concentrators in the prior art can achieve a beam having an irradiance which is greater than the radiance of the source, by employing media having a ratio of refractive indices greater than one.
The limit of the concentration ratio of a non-imaging light concentrating system is determined by the second law of thermodynamics. In the prior art, the limit of the concentration obtainable in a non-imaging system may be estimated using the generalization of the Lagrange invariant for small displacements in position and direction of the incoming ray. Referring to FIG. 1C, an incoming ray is specified in two dimensions by dx, dy, dL and dM. There is a corresponding displacement of the emerging ray specified by primed symbols. Thus, the generalization of the Lagrange invariant asserts that EQU n.sup.2 dx dy dL dM=n'.sup.2 dx'dy'dL'dM', (5)
where L, M and L', M' are direction cosines defining the direction of ray segments in the entry and exit space coordinate systems.
If the system is lossless for the ray in question, this relation maintains that radiance is conserved or not increased along the ray. These limits agree with the predictions of thermodynamics. The statement "radiance is conserved along a ray" must be qualified to exclude certain devices such as dichroic or polarizing beam-combiners, since by combining beams of different frequencies or polarizations, radiance could be increased. However, typically in the prior art, beam-combiners are not used as components of non-imaging light concentrating systems.
When not restricted to small angles, the concentration ratio of a concentrator such as shown in FIG. 1D, having plane entry and exit apertures of areas A and A', respectively, where the entry aperture is filled with rays incident at all angles from 0 to .theta. degrees to the normal and all rays get through the concentrator to emerge from the exit aperture at, say, angles up to .theta.' to the normal, is given by ##EQU2## which is obtained by integrating both sides of Equation (5) over area and solid angle. The maximum possible concentration is obtained when the exit aperture is filled by rays up to .pi./2 to the normal, so the maximum possible concentration ratio is ##EQU3##
Equation (7) is the fundamental relationship of prior art non-imaging optics. It sets general limits on the expected design performance of prior art non-imaging light concentrating systems.
So-called two-dimensional (2D) systems form an important special case of such systems. All of the optical components of these systems are cylindrical having parallel generators, and the entry and exit apertures are parallel-sided slots. It then follows that if end effects are neglected or, as in practice, are canceled optically, the maximum possible concentration ratio is ##EQU4##
Equations (70 and (8), obtained by integrating the differential quantities on either side of Equation (5), which are also known as etendue and throughput, set limits to the designs of prior art concentrators. Typically, it has been possible to reach these limits by suitable designs of 2D non-imaging systems and to approach them asymptotically with three-dimensional, or 3D, systems. Concentrators which fulfill Equations (7) and (8) are called ideal concentrators in the prior art.
Historically, two-dimensional systems using mirrors, rather than refracting elements, were the first to be designed as theoretically perfect concentrators. Refracting elements have so far played more of an auxiliary role in non-imaging concentrators, primarily as convenient tools for making a system more compact or for facilitating the use of total internal reflection.
In the prior art, there are two accepted methods for designing nonimaging light concentrating systems. The first is called the edge-ray method. This design method results in trough-shaped concentrators for two-dimensional (2D) designs, and in cone-shaped concentrators for three-dimensional (3D) designs. For either case, the resulting shape is called a Compound Parabolic Concentrator (hereafter also CPC). FIG. 1D shows the CPC in section, comprising two concave reflectors 10 and 11, each a section of a parabola, but the two are not parts of the same parabola. Starting from the exit aperture A'B', the parabola AA' has its focus at B' and its axis at an angle .theta. to the concentrator axis, as shown. The other side is drawn similarly. The parabolas end at A and A', where their tangents are parallel to the concentrator axis.
With continuing reference to FIG. 1D, the concentration ratio of this system AB/A'B' is 1/sin.theta.. All rays entering at angles up to .theta. get through the exit aperture and no rays at angles greater than .theta. get through. Thus, if the entry aperture is filled by rays up to the angle .+-..theta., the exit aperture must be filled by rays up to .+-..pi./2 in the plane of the diagram. According to the prior art, a plane absorber placed across the exit aperture would receive the maximum possible flux density that could be collected with the entry aperture AB. For a detailed description of the mode of operation of this system, see Winston R., Bassett, I. M., and Welford, W. T., "Nonimaging Optics for Flux Concentration", in Progress in Optics, Vol. 27, pages 161-226, 1989.
In such non-imaging systems, the radiation is supposedly coming from a Lambertian source at infinity subtending the angle .+-..theta., but no image is formed in the exit aperture space. Some rays go through the concentrator after one reflection, some after no reflections, and a sizeable proportion of the transmitted phase space volume is involved in two or more reflections. However, the multiply-reflected rays, to the extent they occur, are intentionally ignored in the design of prior art non-imaging systems. In an image-forming optical system, all the rays which form the image meet each reflecting or refracting surface the same number of times, usually only once, and in refracting systems at least there is generally a well-defined paraxial region in which the image formation is substantially perfect.
It can also be seen from ray traces that the rays which enter at angles less than .theta. follow a variety of paths to get through. The rays at the extreme angle .+-..theta. emerge just grazing the edge of the exit aperture. Thus, the extreme rays, i.e., rays at the maximum entry angle, should just get through an exit aperture of the right size for the planned concentration ratio. Thus, according to the so-called edge-ray principle, extreme rays at the entry aperture should also be extreme rays at the exit aperture. In its more general form, it applies also to sources at a finite distance.
It is important to note that the edge-ray design principle is used to design and implement light concentrating systems in which only rays that are reflected no more than once are considered. While this principle is a heuristic, it is considered helpful in designing both 2D and 3D systems. Referring again to the CPC of FIG. 1D, the extreme entry rays are those at .+-..theta. and the extreme exit rays are those emerging at A' and B'. All light rays entering the concentrator at the maximum design acceptance angle are directed, after one reflection at most, to the rim of the exit aperture. The remaining rays at intermediate angles are reflected within the exit aperture itself. Thus, an application of the principle yields the two parabolic profiles directly.
The second design technique is called the geometric-flux approach. (See Roland Winston, "Nonimaging Optics", Scientific American, pages 76-81, March, 1991.) In this approach, the aggregate of optical rays, i.e. elementary pencils, traverse an optical system in a way analogous to fluid flow. The rays traverse an abstract region called phase space; the space of ray positions and ray geometries. A quantity called geometric flux is constructed from the posited ray positions and directions. The concentrator is designed for a given application in a way that conserves the geometric flux or leaves it undisturbed.
In the example given by Winston to demonstrate the geometric-flux method, a flexible sheet of high-reflecting film is rolled to form a cone, shiny side in, so that a small aperture is left at the tip. The hole is positioned on a round object, such as a ball, and the sides of the cone are adjusted until the entire ball appears to be visible to the person looking through the large end, i.e. large aperture, of the cone. At this point the reflecting cone, i.e. concentrator, does not disturb the geometric vector flux associated with the ball. The lines of flow emanating from the ball are radial because of symmetry, and the cone simply follows those lines. As a consequence, the entire ball appears to be visible, even though all but a small part is hidden from direct view of the eye.
The cone reflects rays of light from the small patch of the ball associated with the small aperture so that the entire ball seems to come into view at the large aperture of the cone. In reverse, rays coming into the cone and heading toward the edge of the ball will be reflected to the small aperture. In other words, rays that would otherwise strike the surface of the ball pass instead through the hole and the light is concentrated.
In most applications, light is concentrated on a flat rather than a spherical surface. The flat surface solution is more complicated, but the basic principles are unchanged. Each flux flow line becomes a hyperbola, and hence the concentrator must be designed with hyperbolic walls. When such a concentrator is placed at the focus of a telescope or a solar furnace, for instance, the instrument appears to have a large target area for incoming light and once again the light is concentrated.
High concentrations of light attained with non-imaging systems are useful in a variety of fields, ranging from high energy physics to solar energy. Non-imaging concentrators collecting solar radiation have been experimentally demonstrated to exceed the radiance level of the sun by as much as fifteen percent (15%). See Cooke, D., Gleckman, P., Krebs, H., O'Gallagher, J., Sagie, D., and Winston R., "Sunlight Brighter than the Sun", Nature, Vol. 346, No. 6287, page 802, Aug. 30, 1990. A concentrator operating in reverse is a projector and can be used as an illumination system.
The prior art design techniques all result in designs which are essentially cone-like or which possess reflecting surface profiles which are close derivatives of parabolic functions. All of the aforementioned designs have common reflector geometries in which the aperture varies monotonically along the primary optical axis. Furthermore, fundamentally these designs intentionally ignore rays having more than a single reflection. Multiply-reflected rays will occur to some extent in the operation of such systems, those rays have not been accounted for in the design of those systems.
Existing designs of light concentrating systems are relatively bulky because they require a relatively large ratio of length to diameter. In addition, in order to exceed the radiance level of the source at the target, light concentrating systems in the prior art must employ optical media having a refractive index greater than one which tends to make them heavy, bulky and expensive. Moreover, the designs of the prior art only perform well for disc-like sources of illumination; consequently, their applicability to a generalized sources or receiver shapes is limited and their performance is severely degraded.